Thermal Amplitudes in DLCQ Superstrings on PP-Waves

We calculate the thermal partition function of DLCQ superstring on the maximally supersymmetric pp-wave background, which is realized as the Penrose limit of orbifolded $AdS_5\times S^5$ and known to be dual to the $\mathcal{N}=2$ ``large'' quiver gauge theory as shown by S. Mukhi, M. Rangamani and E. Verlinde, hep-th/0204147. Making use of the path-integral technique, we derive the manifestly modular invariant expression and show the equivalence with the free energy of second quantized free superstring on this background. The ``virtual strings'' wound around the temporal circle play essential roles for realizing the modular invariance and for a simple analysis on the Hagedorn temperature. We also calculate the thermal one-loop amplitudes of open strings under the various backgrounds of the supersymmetric time-like and Euclidean D-branes, and confirm the existence of correct open-closed string duality. Furthermore, we extend these thermodynamical analysis to the 6-dimensional DLCQ pp-waves with general RR and NSNS flux. These superstring vacua are similarly derived from the supersymmetric (half SUSY) and non-supersymmetric orbifolds of $AdS_3 \times S^3 \times M^4$ ($M^4 =T^4$ or $K3$) by the appropriate Penrose limits, giving rise to the SUSY enhancement as in the case of orbifolded $AdS_5 \times S^5$.